Prove $2^{135}+3^{133}<4^{108}$

How can we prove the following inequality?

$$2^{135}+3^{133}<4^{108}$$

First,

$$2^{135}=2^7\cdot2^{128}=2^7\cdot(2^8)^{16}<(2^8)^{17}=256^{17}\;.$$

Next,

$$3^{133}=3^3\cdot3^{130}=3^3\cdot(3^5)^{26}=3^3(243)^{26}<3^3(256)^{26}=27\cdot256^{26}\;.$$

Thus,

$$2^{135}+3^{133}<256^{17}+27\cdot256^{26}=(1+26\cdot256^9)256^{17}<256^{10}\cdot256^{17}=256^{27}=4^{108}\;.$$

While there are several more clever answers up there, I couldn't resist posting this answer.

2^135 =                          43556142965880123323311949751266331066368
3^133 =   2865014852390475710679572105323242035759805416923029389510561523
4^108 = 105312291668557186697918027683670432318895095400549111254310977536

So even by eye, you can confirm that $2^{135}+3^{133}<4^{108}$.

Using $3^5 < 2^8$ we have $3^{130}<2^{208}$.

$$2^{135}+3^{133}< 2^{135}+3^{3}2^{208}<2^{208}+3^{3}2^{208}=(1+27)4^{104}<4^4 \cdot 4^{104} $$

Note that $2^6<3^4$, so $$2^{135}=2^{129}2^6<2^{129}3^4<3^{129}3^4=3^{133}.$$ Therefore $$2^{135}+3^{133}<3^{133}+3^{133}=2\cdot 3^{133}<3\cdot 3^{133}=3^{134}.$$

Next note that $3^5<2^8$, so $$3^{134}<2^{134\cdot (8/5)}<2^{215}<2^{216}=4^{108}.$$

Hint: log$_2(3)\approx 1.5850$… Consider writing $3^{133}$ as a power of $2$ and factoring.